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The Binomial Theorem

Remember that a binomial is a polynomial with two terms

a + b

x^2 + 1

2y - x

Let's look at the a + b guy for a bit...  Specifically, powers of a + b:

( a + b ) = 1 ... ( a + b )^( 1 ) = a + b ... ( a + b )^( 2 ) = ( a + b )( a + b ) = a^( 2 ) + 2ab + b^( 2 )

 

Here's where the work starts!

( a + b )^( 3 ) = ( a + b )( a + b )^( 2 ) = ( a + b )( a^( 2 ) + 2ab + b^( 2 ) ) = a^( 3 ) + 2a^( 2 )b + ab^( 2 ) + a^( 2 )b + 2ab^( 2 ) + b^( 3 ) = a^( 3 ) + 3a^( 2 )b + 3ab^( 2 ) + b^( 3 )

 

( a + b )^( 4 ) = ( a + b )( a + b )^( 3 ) = ( a + b )( a^( 3 ) + 3a^( 2 )b + 3ab^( 2 ) + b^( 3 ) ) = a^( 4 ) + 3a^( 3 )b + 3a^( 2 )b^( 2 ) + ab^( 3 ) + a^( 3 )b + 3a^( 2 )b^( 2 ) + 3ab^( 3 ) + b^( 4 ) ) = a^( 4 ) + 4a^( 3 )b + 6a^( 2 )b^( 2 ) = 4ab^( 3 ) + b^( 4 )